
theorem Th127:
  for G1, G2 being _Graph, F being non empty PGraphMapping of G1, G2,
    W1 being F-defined Walk of G1
  holds
    F_V.(W1.first()) = (F.:W1).first() & F_V.(W1.last()) = (F.:W1).last()
proof
  let G1, G2 be _Graph, F be non empty PGraphMapping of G1, G2;
  let W1 be F-defined Walk of G1;
  set n = len W1.vertexSeq();
  1 <= n by GLIB_001:67;
  then A1: 1 in dom W1.vertexSeq() & n in dom W1.vertexSeq() by FINSEQ_3:25;
  W1.vertices() = rng W1.vertexSeq() by GLIB_001:def 16;
  then A2: rng W1.vertexSeq() c= dom F_V by Def35;
  then reconsider p = F_V * W1.vertexSeq() as FinSequence by FINSEQ_1:16;
  A3: n = len p by A2, FINSEQ_2:29
    .= len (F.:W1).vertexSeq() by Def37;
  thus F_V.(W1.first()) = F_V.(W1.vertexSeq().1) by GLIB_001:71
    .= (F_V * W1.vertexSeq()).1 by A1, FUNCT_1:13
    .= (F.:W1).vertexSeq().1 by Def37
    .= (F.:W1).first() by GLIB_001:71;
  thus F_V.(W1.last()) = F_V.(W1.vertexSeq().n) by GLIB_001:71
    .= (F_V * W1.vertexSeq()).n by A1, FUNCT_1:13
    .= (F.:W1).vertexSeq().n by Def37
    .= (F.:W1).last() by A3, GLIB_001:71;
end;
